# Normalizing Flows for Interventional Density Estimation

arXiv: https://arxiv.org/abs/2209.06203 · HTML: https://mlsanity.com/paper/2209.06203/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 96 references that are in the map.
- First posted on arXiv 13 Sep 2022.

## Builds on (most cited first, 30 of 43)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests](https://mlsanity.com/paper/1510.04342/) (2015)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Meta-learners for Estimating Heterogeneous Treatment Effects using Machine Learning](https://mlsanity.com/paper/1706.03461/) (2017)
- [Learning Representations for Counterfactual Inference](https://mlsanity.com/paper/1605.03661/) (2016)
- [Adapting Neural Networks for the Estimation of Treatment Effects](https://mlsanity.com/paper/1906.02120/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Bayesian Inference of Individualized Treatment Effects using Multi-task Gaussian Processes](https://mlsanity.com/paper/1704.02801/) (2017)
- [Nonparametric Estimation of Heterogeneous Treatment Effects: From Theory to Learning Algorithms](https://mlsanity.com/paper/2101.10943/) (2021)
- [Semiparametric doubly robust targeted double machine learning: a review](https://mlsanity.com/paper/2203.06469/) (2022)
- [Automated versus do-it-yourself methods for causal inference: Lessons learned from a data analysis competition](https://mlsanity.com/paper/1707.02641/) (2017)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Learning Counterfactual Representations for Estimating Individual Dose-Response Curves](https://mlsanity.com/paper/1902.00981/) (2019)
- [Estimating the Effects of Continuous-valued Interventions using Generative Adversarial Networks](https://mlsanity.com/paper/2002.12326/) (2020)
- [VCNet and Functional Targeted Regularization For Learning Causal Effects of Continuous Treatments](https://mlsanity.com/paper/2103.07861/) (2021)
- [Identifying Causal-Effect Inference Failure with Uncertainty-Aware Models](https://mlsanity.com/paper/2007.00163/) (2020)
- [Conditional Density Estimation with Bayesian Normalising Flows](https://mlsanity.com/paper/1802.04908/) (2018)
- [Quantifying Ignorance in Individual-Level Causal-Effect Estimates under Hidden Confounding](https://mlsanity.com/paper/2103.04850/) (2021)
- [Semiparametric counterfactual density estimation](https://mlsanity.com/paper/2102.12034/) (2021)
- [Invertible Generative Modeling using Linear Rational Splines](https://mlsanity.com/paper/2001.05168/) (2020)
- [Graphical Normalizing Flows](https://mlsanity.com/paper/2006.02548/) (2020)
- [Benchmarking Framework for Performance-Evaluation of Causal Inference Analysis](https://mlsanity.com/paper/1802.05046/) (2018)
- [Counterfactual Mean Embeddings](https://mlsanity.com/paper/1805.08845/) (2018)
- [Combining Observational and Randomized Data for Estimating Heterogeneous Treatment Effects](https://mlsanity.com/paper/2202.12891/) (2022)

## Built on by (most cited first, 21 of 21)

- [Causal machine learning for predicting treatment outcomes](https://mlsanity.com/paper/2410.08770/) (2024)
- [Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation](https://mlsanity.com/paper/2310.17463/) (2023)
- [Sharp Bounds for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2305.16988/) (2023)
- [Counterfactual Generative Models for Time-Varying Treatments](https://mlsanity.com/paper/2305.15742/) (2023)
- [A Neural Framework for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2311.16026/) (2023)
- [Reliable Off-Policy Learning for Dosage Combinations](https://mlsanity.com/paper/2305.19742/) (2023)
- [Structured Neural Networks for Density Estimation and Causal Inference](https://mlsanity.com/paper/2311.02221/) (2023)
- [IGC-Net for conditional average potential outcome estimation over time](https://mlsanity.com/paper/2405.21012/) (2024)
- [Stabilized Neural Prediction of Potential Outcomes in Continuous Time](https://mlsanity.com/paper/2410.03514/) (2024)
- [Scalable Counterfactual Distribution Estimation in Multivariate Causal Models](https://mlsanity.com/paper/2311.00927/) (2023)
- [Deep Autoregressive Models as Causal Inference Engines](https://mlsanity.com/paper/2409.18581/) (2024)
- [chiSPN: Characteristic Interventional Sum-Product Networks for Causal Inference in Hybrid Domains](https://mlsanity.com/paper/2408.07545/) (2024)
- [Flow-based Generative Modeling of Potential Outcomes and Counterfactuals](https://mlsanity.com/paper/2505.16051/) (2025)
- [A Diffusion-Based Method for Learning the Multi-Outcome Distribution of Medical Treatments](https://mlsanity.com/paper/2506.01533/) (2025)
- [PrivATE: Differentially Private Confidence Intervals for Average Treatment Effects](https://mlsanity.com/paper/2505.21641/) (2025)
- [ExOSITO: Explainable Off-Policy Learning with Side Information for Intensive Care Unit Blood Test Orders](https://mlsanity.com/paper/2504.17277/) (2025)
- [GDR-learners: Orthogonal Learning of Generative Models for Potential Outcomes](https://mlsanity.com/paper/2509.22953/) (2025)
- [Overlap-Adaptive Regularization for Conditional Average Treatment Effect Estimation](https://mlsanity.com/paper/2509.24962/) (2025)
- [Debiased Counterfactual Generation via Flow Matching from Observations](https://mlsanity.com/paper/2605.07665/) (2026)
- [Benchmarking Counterfactual Prediction in Epidemic Time Series with Time-Varying Interventions](https://mlsanity.com/paper/2606.05692/) (2026)
- [Recovering Weak Signals with Normalizing Flows](https://mlsanity.com/paper/2609.06382/) (2026)

## Abstract (arXiv)

Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of potential outcomes (e.g., average treatment effect). However, such quantities do not capture the full information about the distribution of potential outcomes. In this work, we estimate the density of potential outcomes after interventions from observational data. For this, we propose a novel, fully-parametric deep learning method called Interventional Normalizing Flows. Specifically, we combine two normalizing flows, namely (i) a nuisance flow for estimating nuisance parameters and (ii) a target flow for parametric estimation of the density of potential outcomes. We further develop a tractable optimization objective based on a one-step bias correction for efficient and doubly robust estimation of the target flow parameters. As a result, our Interventional Normalizing Flows offer a properly normalized density estimator. Across various experiments, we demonstrate that our Interventional Normalizing Flows are expressive and highly effective, and scale well with both sample size and high-dimensional confounding. To the best of our knowledge, our Interventional Normalizing Flows are the first proper fully-parametric, deep learning method for density estimation of potential outcomes.